Affine group representation formalism for four-dimensional, Lorentzian, quantum gravity

Abstract

Within the context of the Ashtekar variables, the Hamiltonian constraint of four-dimensional pure general relativity with cosmological constant, Λ, is re-expressed as an affine algebra with the commutator of the imaginary part of the Chern–Simons functional, Q, and the positive-definite volume element. This demonstrates that the affine algebra quantization program of Klauder can indeed be applicable to the full Lorentzian signature theory of quantum gravity with non-vanishing cosmological constant, and it facilitates the construction of solutions to all of the constraints. Unitary, irreducible representations of the affine group exhibit a natural Hilbert space structure, and coherent states and other physical states can be generated from a fiducial state. It is also intriguing that formulation of the Hamiltonian constraint or the Wheeler–DeWitt equation as an affine algebra requires a non-vanishing cosmological constant, and a fundamental uncertainty relation of the form (wherein V is the total volume) may apply to all physical states of quantum gravity.